"Gift from ~" refers to who gave the gift. So, "A gift from John" means that John gave the gift.

"Gift of ~" refers to that which was given. "A gift of John" means that John was given as a gift. Between three consenting adults, there may not be anything wrong with that, but in most cases it would get you a raised eyebrow.

You often see this construction around holidays in advertisements. "Give the gift of warmth," for example, might be used to sell sweaters. Or charitable donations - "She gave a gift of $50 to her church."

@mshades - On the contrary, you were quite right to use "between". It's nonsense that "between" can only be used for two, and anyone who gives you a smacking for it has only learnt half a rule. When we see things as a group, yes we use "among", but when we see them as individuals, we tend to use "between". This is from the OED:

"It [between] is still the only word available to express the relation of a thing to many surrounding things severally and individually, among expressing a relation to them collectively and vaguely"

"The Republic of Poland, a country in Central Europe, lies between Germany to the west, the Czech Republic and Slovakia to the south, Ukraine and Belarus to the east, and the Baltic Sea, Lithuania and Russia to the north." - UCLA Center for European and Eurasian Studies